OR

The polar equation of a curve \(C\) is \(r=a(1+\cos\theta)\), for \(0\leq\theta\lt 2\pi\), where \(a\) is a positive constant.

(i) Sketch \(C\).

(ii) Show that the Cartesian equation of \(C\) is

\(x^2+y^2=a\left(x+\sqrt{x^2+y^2}\right).\)

(iii) Find the area of the sector of \(C\) between \(\theta=0\) and \(\theta=\frac{\pi}{3}\).

(iv) Find the arc length of \(C\) between \(\theta=0\) and \(\theta=\frac{\pi}{3}\).